The unifiedΩ-series of the Gauss and Bailey2F1(1/2)-sums will be investigated by utilizing asymptotic methods and the modified Abel lemma on summation by parts.Several remarkable transformation theorems for theΩ-ser...The unifiedΩ-series of the Gauss and Bailey2F1(1/2)-sums will be investigated by utilizing asymptotic methods and the modified Abel lemma on summation by parts.Several remarkable transformation theorems for theΩ-series will be proved whose particular cases turn out to be strange evaluations of nonterminating hypergeometric series and infinite series identities of Ramanujan-type,including a couple of beautiful expressions forπand the Catalan constant discovered by Guillera(2008).展开更多
Recently,scrambling index and competition index are widely applied to stochastic matrices and food webs. By analyzing the relationship of scrambling index and 2-competition index,n-「d/2」+ 1 was proved to be an upper...Recently,scrambling index and competition index are widely applied to stochastic matrices and food webs. By analyzing the relationship of scrambling index and 2-competition index,n-「d/2」+ 1 was proved to be an upper bound of the 2-competition2 index of a primitive digraph with exact d loops in this article.Moreover,the maximum index problem and the index set problem for the 2-competition index of primitive digraphs with minimally strong digraphs were settled.展开更多
The 2-sum of two digraphs and , denoted , is the digraph obtained from the disjoint union of and by identifying an arc in with an arc in . A digraph D is supereulerian if D contains a spanning eulerian subdigraph. It ...The 2-sum of two digraphs and , denoted , is the digraph obtained from the disjoint union of and by identifying an arc in with an arc in . A digraph D is supereulerian if D contains a spanning eulerian subdigraph. It has been noted that the 2-sum of two supereulerian (or even hamiltonian) digraphs may not be supereulerian. We obtain several sufficient conditions on and for to be supereulerian. In particular, we show that if and are symmetrically connected or partially symmetric, then is supereulerian.展开更多
A ring is said to satisfy the strong 2-sum property if every element is a sum of two commuting units.In this note,we present some sufficient or necessary conditions for the matrix ring over a commutative local ring to...A ring is said to satisfy the strong 2-sum property if every element is a sum of two commuting units.In this note,we present some sufficient or necessary conditions for the matrix ring over a commutative local ring to have the strong 2-sum property.展开更多
It is known that the study of the qualitative properties of a matrix A (which depend only on the sign pattern of A) can be turned into the study of the graph theoretical properties of the signed digraph S(A). The unde...It is known that the study of the qualitative properties of a matrix A (which depend only on the sign pattern of A) can be turned into the study of the graph theoretical properties of the signed digraph S(A). The underlying digraph of the signed digraph of a strong sign nonsingular matrix (abbreviated S NS matrix) with a negative main diagonal is called an S NS digraph. In the study of S NS digraphs, the minimal forbidden configuration (or MFC for short) plays an important role. Three (classes of) MFS's were constructed by Thomassen, Brualdi and Shader, and Shao. In this paper, we show that a digraph D is an S2NS digraph if and only if its 'cycle linear system' is solvable. This simplifies a parallel result obtained by Shao and Hu. As an application of the result, a graph theoretical characterization for a digraph to be an S NS digraph is given. At the end of the paper, we construct infinitely many new MFCs to show that for each even number k(k>0), there are basic MFCs with fc terminal components (here, with no loss of generality, we assume that the number of the initial components of a digraph is no less than that of its terminal components throughout the following).展开更多
文摘The unifiedΩ-series of the Gauss and Bailey2F1(1/2)-sums will be investigated by utilizing asymptotic methods and the modified Abel lemma on summation by parts.Several remarkable transformation theorems for theΩ-series will be proved whose particular cases turn out to be strange evaluations of nonterminating hypergeometric series and infinite series identities of Ramanujan-type,including a couple of beautiful expressions forπand the Catalan constant discovered by Guillera(2008).
基金National Natural Science Foundations of China(No.11272100,No.50865001)
文摘Recently,scrambling index and competition index are widely applied to stochastic matrices and food webs. By analyzing the relationship of scrambling index and 2-competition index,n-「d/2」+ 1 was proved to be an upper bound of the 2-competition2 index of a primitive digraph with exact d loops in this article.Moreover,the maximum index problem and the index set problem for the 2-competition index of primitive digraphs with minimally strong digraphs were settled.
文摘The 2-sum of two digraphs and , denoted , is the digraph obtained from the disjoint union of and by identifying an arc in with an arc in . A digraph D is supereulerian if D contains a spanning eulerian subdigraph. It has been noted that the 2-sum of two supereulerian (or even hamiltonian) digraphs may not be supereulerian. We obtain several sufficient conditions on and for to be supereulerian. In particular, we show that if and are symmetrically connected or partially symmetric, then is supereulerian.
基金This research was supported by the Natural Science Foundation of China(grants 11661014,11661013,11961050)the Guangxi Natural Science Foundation(grant no.2016GXNSFDA380017)a Discovery Grant from NSERC of Canada(grant no.RGPIN-2016-04706).
文摘A ring is said to satisfy the strong 2-sum property if every element is a sum of two commuting units.In this note,we present some sufficient or necessary conditions for the matrix ring over a commutative local ring to have the strong 2-sum property.
基金Supported by the Doctoral Fund of the Xiamen University (No.Y01007).
文摘It is known that the study of the qualitative properties of a matrix A (which depend only on the sign pattern of A) can be turned into the study of the graph theoretical properties of the signed digraph S(A). The underlying digraph of the signed digraph of a strong sign nonsingular matrix (abbreviated S NS matrix) with a negative main diagonal is called an S NS digraph. In the study of S NS digraphs, the minimal forbidden configuration (or MFC for short) plays an important role. Three (classes of) MFS's were constructed by Thomassen, Brualdi and Shader, and Shao. In this paper, we show that a digraph D is an S2NS digraph if and only if its 'cycle linear system' is solvable. This simplifies a parallel result obtained by Shao and Hu. As an application of the result, a graph theoretical characterization for a digraph to be an S NS digraph is given. At the end of the paper, we construct infinitely many new MFCs to show that for each even number k(k>0), there are basic MFCs with fc terminal components (here, with no loss of generality, we assume that the number of the initial components of a digraph is no less than that of its terminal components throughout the following).
